Abundance of Dirichlet-improvable pairs with respect to arbitrary norms

TL;DR

This paper extends the concept of Dirichlet spectrum to arbitrary norms in two dimensions, showing that the set of Dirichlet improvable pairs is large and contains badly approximable pairs, with implications for the structure of the spectrum.

Contribution

It generalizes the Dirichlet spectrum to all norms on 2 and proves the set of Dirichlet improvable pairs is hyperplane absolute winning, not isolated in the spectrum.

Findings

The set of Dirichlet improvable pairs contains badly approximable pairs.

The Dirichlet spectrum's top point is not isolated.

The set of improvable pairs is hyperplane absolute winning.

Abstract

In a recent paper of Akhunzhanov and Shatskov the two-dimensional Dirichlet spectrum with respect to Euclidean norm was defined. We consider an analogous definition for arbitrary norms on $\mathbb{R}^2$ and prove that, for each such norm, the set of Dirichlet improvable pairs contains the set of badly approximable pairs, hence is hyperplane absolute winning. To prove this we make a careful study of some classical results in the geometry of numbers due to Chalk--Rogers and Mahler to establish a Haj\'{o}s--Minkowski type result for the critical locus of a cylinder. As a corollary, using a recent result of the first named author with Mirzadeh, we conclude that for any norm on $\mathbb{R}^2$ the top of the Dirichlet spectrum is not an isolated point.